91Ó°ÊÓ

1. Speed of sound in metal is$\mathrm{v}$ .
2. Speed of sound in metal is${\mathrm{v}}_{\mathrm{m}}$.
3. Length of tube is$\mathrm{L}$ .

Using the formula of velocity, we can derive the equation for the time interval between the arrivals of the two sounds at the listener’s ear if v is the speed of sound in air. Using this derivation, we can find the length of the pipe.

Formula:

The velocity of the sound wave,

$\mathbf{v}\mathbf{=}\mathbf{d}\mathbf{/}\mathbf{t}$ …(¾±)

Using equation (i), we can the time taken by the body to be:

$\mathrm{t}=\frac{\mathrm{L}}{\mathrm{v}}$

$âˆ\mu \mathrm{d}=\mathrm{L}$

Time t for sound to travel in the metal is${\mathrm{t}}_{\mathrm{m}}=\mathrm{L}/{\mathrm{v}}_{\mathrm{m}}$

Time t for sound to travel in air is${\mathrm{t}}_{\mathrm{a}}=\mathrm{L}/\mathrm{v}$

So, the time interval between these sounds can be given as:

$\begin{array}{c}\mathrm{Δ³Ù}={\mathrm{t}}_{\mathrm{a}}−{\mathrm{t}}_{\mathrm{m}}\\ \mathrm{Δ³Ù}=\frac{\mathrm{L}}{\mathrm{v}}−\frac{\mathrm{L}}{{\mathrm{v}}_{\mathrm{m}}}\end{array}$

$\mathrm{Δ³Ù}=\frac{\mathrm{L}({\mathrm{v}}_{\mathrm{m}}−\mathrm{v})}{{\mathrm{v}}_{\mathrm{m}}\mathrm{v}}$ …(²¹)

Therefore, the time interval between the arrivals of the two sounds at the listener’s ear if $\mathrm{v}$is the speed of sound in air is $\frac{\mathrm{L}({\mathrm{v}}_{\mathrm{m}}−\mathrm{v})}{{\mathrm{v}}_{\mathrm{m}}\mathrm{v}}$.

We have the length of a material using equation (a) as:

$\mathrm{L}=\frac{\mathrm{Δ³Ù}}{\left(\frac{1}{\mathrm{v}}−\frac{1}{{\mathrm{v}}_{\mathrm{m}}}\right)}$

For metal that is for steel pipe, ${\mathrm{v}}_{\mathrm{m}}=5941\text{″¾}/\text{s}$and as $\mathrm{v}=343\text{″¾}/\text{s},\mathrm{Δ³Ù}=1.00\text{ s}$.

Hence, the length value is given as:

$\begin{array}{c}\mathrm{L}=\frac{1.00\text{ s}}{\left(\frac{1}{343\text{″¾}/\text{s}}−\frac{1}{5941\text{″¾}/\text{s}}\right)}\\ =364\text{″¾}\end{array}$

Therefore, the length $L$, if $\mathrm{Δ³Ù}=1.00\mathrm{s}$and the metal is steel is$364\text{″¾}$ .